Calculus of Variations and Geometric Measure Theory

Vector fields, currents and rectifiable curves in metric spaces

Eugene Stepanov (St. Petersburg Branch of the Steklov Research Institute of Mathematics of the Russian Academy of Sciences)

created by magnani on 06 Dec 2012

12 dec 2012 -- 17:00   [open in google calendar]

Sala Seminari, Department of Mathematics, Pisa University


There are two ways to view (smooth) vector fields over smooth manifolds, namely

(1) as derivations over an algebra of smooth functions, (2) as ``vectors attached to points'', i.e. directions of curves.

Both notions are in fact equivalent, and while (1) is more abstract, it is easily extended to arbitrary metric spaces (with smooth functions substituted by Lipschitz ones), thus leading to the identification ``vector fields = one-dimensional currents''. It will be shown that even in such a generality vector fields corresponding to normal currents have an underlying structure, provided by rectifiable curves (i.e. they can be viewed, in a sense, like (2)). Several consequences of this fact will be discussed.